The algebra of enumeration operators

We study the algebra of all operators of enumeration ($e$-operators) of the form $\mathscr{E}=\langle E,I,*\rangle$. Here $E,I,*$ stand respectively for the set of all $e$-operators, the identity operator and the binary operation of superposition on $E$. We prove the existence of $n$-element bases ($n\ge2$), the continuality of the family of maximal subalgebras in $\mathscr{E}$ and show that $\mathscr{E}$ is not finitely presented.